Dynamics 14th edition by r c hibbeler chapter 19

Show that the angular momentum of the body computed about the instantaneous center of zero velocity IC can be expressed as , where represents the body’s moment of inertia computed about

Q.E.D.

rP>G = k

2 G

rG>OHowever, yG = vrG>O or rG>O = yvG

rP>G = k

2 G

The rigid body (slab) has a mass m and rotates with an

angular velocity about an axis passing through the fixed

point O Show that the momenta of all the particles

composing the body can be represented by a single vector

having a magnitude and acting through point P, called

the center of percussion, which lies at a distance

from the mass center G Here is the

radius of gyration of the body, computed about an axis

perpendicular to the plane of motion and passing through G.

At a given instant, the body has a linear momentum

and an angular momentum computed

about its mass center Show that the angular momentum of

the body computed about the instantaneous center of zero

velocity IC can be expressed as , where

represents the body’s moment of inertia computed about

the instantaneous axis of zero velocity As shown, the IC is

located at a distance rG>ICaway from the mass center G.

Since , the linear momentum Hence the angular momentum

about any point P is

HP= IGv

L = myG = 0

yG = 0

Show that if a slab is rotating about a fixed axis perpendicular

to the slab and passing through its mass center G, the angular

momentum is the same when computed about any other

point P.

P

G

V

Equilibrium Since slipping occurs at brake pad, F f = mk N = 0.3 N

Referring to the FBD the brake’s lever, Fig a,

The 40-kg disk is rotating at V = 100 rad>s When the force

P is applied to the brake as indicated by the graph If the

coefficient of kinetic friction at B is m k = 0.3, determine

the time t needed to stay the disk from rotating Neglect the

thickness of the brake

The impact wrench consists of a slender 1-kg rod AB which

is 580 mm long, and cylindrical end weights at A and B that

each have a diameter of 20 mm and a mass of 1 kg This

which are attached to the lug nut on the wheel of a car If

the rod AB is given an angular velocity of 4 and it

strikes the bracket C on the handle without rebounding,

determine the angular impulse imparted to the lug nut

8 m

8 m

A G B

T A  40 kN

T B  20 kN

The airplane is traveling in a straight line with a speed of

300 km h, when the engines A and B produce a thrust of

and , respectively Determine the

angular velocity of the airplane in The plane has a

mass of 200 Mg, its center of mass is located at G, and its

radius of gyration about G is kG = 15 m

t = 5 s

TB = 20 kN

TA= 40 kN>

SOLUTION

Principle of Angular Impulse and Momentum: The mass moment of inertia of the

Applying the angular impulse and momentum equation about point G,

The double pulley consists of two wheels which are attached

to one another and turn at the same rate The pulley has a

mass of 15 kg and a radius of gyration of If

the block at A has a mass of 40 kg, determine the speed

of the block in 3 s after a constant force of 2 kN is applied

to the rope wrapped around the inner hub of the pulley.The

block is originally at rest

a asyst angular momentumbO

1 + a a syst angular impulsebO

1 ft

1 ft0.8 ft

G

The assembly weighs 10 lb and has a radius of gyration

about its center of mass G The kinetic energy

of the assembly is when it is in the position shown

If it is rolling counterclockwise on the surface without

slipping, determine its linear momentum at this instant

The disk has a weight of 10 lb and is pinned at its center O.

If a vertical force of is applied to the cord wrapped

around its outer rim, determine the angular velocity of the

disk in four seconds starting from rest Neglect the mass of

The 30-kg gear A has a radius of gyration about its center of

mass O of If the 20-kg gear rack B is

subjected to a force of , determine the time

required for the gear to obtain an angular velocity of

, starting from rest.The contact surface between the

gear rack and the horizontal plane is smooth.20 rad>s

Principle of Impulse and Momentum: Applying the linear impulse and momentum

equation along the x axis using the free-body diagram of the gear rack shown in Fig a,

(1)

The mass moment of inertia of the gear about its mass center is

Writing the angular impulse and momentum

equation about point O using the free-body diagram of the gear shown in Fig b,

Principle of Impulse and Momentum: The mass moment inertia of the pulley about its

eahew,4–1.qEgniylpA.s

IOv1 + © 1t2

t 1Modt = Iov2

Io = 12 a32.2 b8 (0.62) = 0.04472 slug#ft2

The pulley has a weight of 8 lb and may be treated as a thin

disk A cord wrapped over its surface is subjected to forces

and Determine the angular velocity

of the pulley when if it starts from rest when

Neglect the mass of the cord

The 40-kg roll of paper rests along the wall where the

coefficient of kinetic friction is mk = 0.2 If a vertical force

of P = 40 N is applied to the paper, determine the angular

velocity of the roll when t = 6 s starting from rest Neglect

the mass of the unraveled paper and take the radius of

13

O B

120 mm

P  40 N

A

Solution

Principle of Impulse and Momentum The mass moment of inertia of the paper roll

about its center is I O = mk O2 = 40(0.082) = 0.256 kg#m2 Since the paper roll is

required to slip at point of contact, F f = mk N = 0.2 N Referring to the FBD of the

paper roll, Fig a,

The slender rod has a mass m and is suspended at its end A

by a cord If the rod receives a horizontal blow giving it an

impulse I at its bottom B, determine the location y of the

point P about which the rod appears to rotate during the

impact

SOLUTION

Principle of Impulse and Momentum:

(a

Kinematics: Point P is the IC.

Using similar triangles,

Ans:

y = 23 l

The rod of length L and mass m lies on a smooth horizontal

surface and is subjected to a force P at its end A as shown

Determine the location d of the point about which the rod

begins to turn, i.e, the point that has zero velocity

A

P

L d

A 4-kg disk A is mounted on arm BC, which has a negligible

mass If a torque of where t is in seconds,

is applied to the arm at C, determine the angular velocity of

BC in 2 s starting from rest Solve the problem assuming

that (a) the disk is set in a smooth bearing at B so that it

rotates with curvilinear translation, (b) the disk is fixed to

the shaft BC, and (c) the disk is given an initial freely

spinning angular velocity of prior to

application of the torque

1.5 ft 1.5 ft

M 300 lb ft

The frame of a tandem drum roller has a weight of 4000 lb

excluding the two rollers Each roller has a weight of 1500 lb

and a radius of gyration about its axle of 1.25 ft If a torque

of is supplied to the rear roller A, determine

the speed of the drum roller 10 s later, starting from rest.M = 300 lb#ft

SOLUTION

Principle of Impulse and Momentum: The mass moments of inertia of the rollers

rollers roll without slipping, Using the free-body

diagrams of the rear roller and front roller, Figs a and b, and the momentum

diagram of the rollers, Fig c,

Principle of Impulse and Momentum: We can eliminate the force F from the

analysis if we apply the principle of impulse and momentum about point A The

mass moment inertia of the wheel about point A is

Applying Eq 19–14, we have

Kinematics: Since the wheel rolls without slipping at point A, the instantaneous

center of zero velocity is located at point A Thus,

The 100-lb wheel has a radius of gyration of

If the upper wire is subjected to a tension of

determine the velocity of the center of the wheel in 3 s,

starting from rest.The coefficient of kinetic friction between

the wheel and the surface is mk = 0.1

The 4-kg slender rod rests on a smooth floor If it is kicked

so as to receive a horizontal impulse at point A

as shown, determine its angular velocity and the speed of its

mass center

I = 8 N#s

2 m 1.75 m

The double pulley consists of two wheels which are

attached to one another and turn at the same rate The

pulley has a mass of 15 kg and a radius of gyration

If the block at A has a mass of 40 kg,

determine the speed of the block in 3 s after a constant

force is applied to the rope wrapped around the

inner hub of the pulley The block is originally at rest

Neglect the mass of the rope

The 100-kg spool is resting on the inclined surface for which

the coefficient of kinetic friction is mk = 0.1 Determine the

angular velocity of the spool when t = 4 s after it is released

from rest The radius of gyration about the mass center is

k G = 0.25 m

30

G A

0.2 m 0.4 m

Solution

Kinematics The IC of the spool is located as shown in Fig a Thus

vG = vr G >IC = v(0.2)

Principle of Impulse and Momentum The mass moment of inertia of the spool

about its mass center is I G = mk G2 = 100(0.252) = 6.25 kg#m2 Since the spool is

required to slip, F f = mk N = 0.1 N Referring to the FBD of the spool, Fig b,

Solving Eqs (1) and (2),

T = 313.59 N

Ans:

v = 18.4 rad>s b

The spool has a weight of 30 lb and a radius of gyration

A cord is wrapped around its inner hub and the

end subjected to a horizontal force Determine the

spool’s angular velocity in 4 s starting from rest Assume

the spool rolls without slipping

P= 5 lb

kO = 0.45 ft

P 5 lb 0.9 ft

5(3) + 49.05 sin 30°(t) - 25.487t = 5vG

b+ mvx1 + aLFxdt = mvx2

The hoop (thin ring) has a mass of 5 kg and is released down

the inclined plane such that it has a backspin

and its center has a velocity as shown If the

coefficient of kinetic friction between the hoop and the

plane is determine how long the hoop rolls before

The 30-kg gear is subjected to a force of , where

tis in seconds Determine the angular velocity of the gear at

, starting from rest Gear rack B is fixed to the

horizontal plane, and the gear’s radius of gyration about its

mass center O is kO = 125 mm

t = 4 s

P = (20t) N

SOLUTION

Kinematics: Referring to Fig a,

Principle of Angular Impulse and Momentum: The mass moment of inertia of

Writing the angular impulse and momentum equation about point A shown in

The 30-lb flywheel A has a radius of gyration about its center

of 4 in Disk B weighs 50 lb and is coupled to the flywheel by

means of a belt which does not slip at its contacting surfaces

If a motor supplies a counterclockwise torque to the

flywheel of , where t is in seconds,

determine the time required for the disk to attain an angular

velocity of 60 rad>sstarting from rest

M = (50t) lb#f t

SOLUTION

Principle of Impulse and Momentum: The mass moment inertia of the flywheel

about point C is The angular velocity of the

flywheel [FBD(a)], we have

(a

(1)

The mass moment inertia of the disk about point D is

Applying Eq 19–14 to the disk [FBD(b)], we have

SOLUTION

Principle of Impulse and Momentum: The mass moment of inertia of the rods

If the shaft is subjected to a torque of ,

where t is in seconds, determine the angular velocity of the

assembly when , starting from rest Rods AB and BC

each have a mass of 9 kg

vB = 1.59 m>s

The double pulley consists of two wheels which are attached

to one another and turn at the same rate The pulley has a

mass of 15 kg and a radius of gyration of

k O = 110 mm If the block at A has a mass of 40 kg and the

container at B has a mass of 85 kg, including its contents,

determine the speed of the container when t = 3 s after it is

released from rest

Solution

The angular velocity of the pulley can be related to the speed of

container B by v = 0.075 =vA 13.333 vB Also the speed of block A

SOLUTION

The number of rollers per unit length is 1/d.

Thus in one second, rollers are contacted

If a roller is brought to full angular speed of in t0seconds, then the moment

of inertia that is effected is

Since the frictional impluse is

The crate has a mass Determine the constant speed it

acquires as it moves down the conveyor The rollers each

have a radius of r, mass m, and are spaced d apart Note that

friction causes each roller to rotate when the crate comes in

contact with it

The turntable T of a record player has a mass of 0.75 kg and

a radius of gyration k z = 125 mm It is turning freely at

vT = 2 rad>s when a 50-g record (thin disk) falls on it

Determine the final angular velocity of the turntable just

after the record stops slipping on the turntable

The 10-g bullet having a velocity of 800 m>s is fired into the

edge of the 5-kg disk as shown Determine the angular

velocity of the disk just after the bullet becomes embedded

into its edge Also, calculate the angle u the disk will swing

when it stops The disk is originally at rest Neglect the mass

of the rod AB.

Conservation of Angular Momentum The mass moment of inertia of the disk about

its mass center is (I G)d = 12 (5)(0.42) = 0.4 kg#m2 Also, (vb)2 = v2125.922 and

(vd)2 = v2(2.4) Referring to the momentum diagram with the embedded bullet,

Fig a,

Σ(H A)1 = Σ(H A)2

M b(vb)1(r b)1 = (I G)d v2+ M d(vd)2(r2) + M b(vb)2(r b)2

0.01(800)(2.4) = 0.4v2 + 53v2(2.4)4(2.4) + 0.013v2125.922 4 125.922

Kinetic Energy Since the system is required to stop finally, T3 = 0 Here

T2 = 12 (I G)dv2+ 12 Md(v d)2 + 12 M b(vb)2

= 12 (0.4)(0.65622) + 1

2(5)[0.6562(2.4)]2+ 12(0.01)30.6562125.92242

= 6.2996 J

Potential Energy Datum is set as indicated on Fig b.

Here f = tan -1 a0.42.4 b =9.4623 Hence

The 10-g bullet having a velocity of 800 m>s is fired into the

edge of the 5-kg disk as shown Determine the angular

velocity of the disk just after the bullet becomes embedded

into its edge Also, calculate the angle u the disk will swing

when it stops The disk is originally at rest The rod AB has a

Conservation of Angular Momentum The mass moments of inertia of the disk and

rod about their respective mass centers are (I G)d = 12 (5)(0.42) = 0.4 kg#m2 and

Kinetic Energy Since the system is required to stop finally, T3= 0 Here

19–31 Continued

Potential Energy Datum is set as indicated on Fig b

Here f = tan-1a0.42.4 b = 9.4623° Hence

y d = 2.4 cos u, y r = cos u, y b = 25.92 cos (u - 9.4623°)

Thus, the gravitational potential energy of the disk, rod and bullet with reference to

The circular disk has a mass m and is suspended at A by the

wire If it receives a horizontal impulse I at its edge  B,

determine the location y of the point P about which the disk

appears to rotate during the impact

Principle of Impulse and Momentum The mass moment of inertia of the disk about

its mass center is I G = 12mr2 = 12ma2

0.65 m0.20 m

The 80-kg man is holding two dumbbells while standing on a

turntable of negligible mass, which turns freely about a

vertical axis When his arms are fully extended, the

turn-table is rotating with an angular velocity of

Determine the angular velocity of the man when he retracts

his arms to the position shown When his arms are fully

extended, approximate each arm as a uniform 6-kg rod

having a length of 650 mm, and his body as a 68-kg solid

cylinder of 400-mm diameter With his arms in the retracted

position, assume the man as an 80-kg solid cylinder of 450-mm

diameter Each dumbbell consists of two 5-kg spheres of

negligible size

0.5 rev>s

SOLUTION

Conservation of Angular Momentum: Since no external angular impulse acts on the

system during the motion, angular momentum about the axis of rotation (z axis) is

conserved The mass moment of inertia of the system when the arms are in the fully